Background to Sound waves and Resonance
What are waves? A wave is a disturbance in both space and time; an oscillation which transfers energy. There are three ways that waves can propagate through space, these are transverse, longitudinal, or surface waves. A transverse wave is arguably the easiest to imagine since we see these most commonly, a prime example being a wave on a string. A transverse wave is where the molecules are oscillating perpendicular to the plane of propagation. This is shown in Figure 1, a wave on the sea, where the waves can be seen moving in time as well as moving water towards the shore. All waves are similar to this, but vary in how they can be described mathematically. For simplicity here, we will only deal with sinusoidal waves. A longitudinal wave is one whose oscillations are parallel to the plane of movement. This occurs as shown in Figure 2 by compressions, areas of high pressure, and rarefactions, areas of low pressure. Many factors affect these waves and will be described in more detail in the Sound Waves section below. The third type of wave propagation, a surface wave, will only be described here conceptually to ensure a complete picture of waves. These are most commonly seen on the surface of the ocean. The particles or molecules in a surface wave move in a circular pattern. As Figure 3 shows, there is the same oscillatory motion as longitudinal and transverse waves, but surface waves are the hardest to describe mathematically. There are two types of wave, electromagnetic and mechanical. Sound waves are mechanical, and this means that the wave requires a medium in which to propagate, such as air or a metal. The particles within the media oscillate and form the propagating wave. Electromagnetic waves, light, on the other hand, do not require a medium to be able to transfer energy. This is why you can see stars, the light from them can travel across space to Earth. Now we have covered the basics of all waves, we can move onto learning more specifically about sound waves. We will describe sound waves and what factors affect their propagation, along with the maths needed to describe these waves. From here you will gain the knowledge required to understand the physics behind the Resonance Tube experiment and how to calculate the speed of sound in air. Sound waves A sound wave is a vibration that propagates as a mechanical wave of pressure and displacement. Unlike light waves, sound waves cannot travel through a vacuum as they require a medium of matter to propagate through such as air or water. Sound waves can also be longitudinal (Figure 2) or transverse (Figure 4), depending on the medium they are propagating though. For example, if the sound wave propagates through air or water, it will take on a longitudinal form and for solids the wave will be transverse in most cases. Generating a sound wave generally involves a vibrating source, such as a stereo speaker. The source causes vibrations in the surrounding medium which, as the source continuously vibrates, propagate away from the source at the speed of sound. If we were to consider a point that is a fixed distance away from the source we would find that the pressure, velocity and displacement of the medium vary in time. Similarly, if we were to freeze time and look at the static wave, the pressure, velocity and displacement of the medium vary in space. One misconception often found is that you may think the particles in the medium, for example air, travel with the wave as it propagates. This is not the case, as the particles oscillate around their equilibrium point (in other words, their average position), which is static, while the wave propagates as if it were superimposed on top of the motion of the particles. This can be seen from Figure 2. The particles act as a 'transporter' for the vibrations. Whilst sound waves can be described by the same simple mathematical laws that govern all waves, how the collective motion of all the particles in the medium and how the properties of the medium itself correspond to the wave is much more complicated. Firstly, there is a complex relationship between the density and pressure of the medium which is affected by temperature. This relationship determines the speed of sound and so, unlike light waves which have constant speed in a vacuum, the speed of sound is dependent on the properties of the medium it propagates through. For example, the speed of sound in dry air at 20 degrees Celsius at 'sea level' atmospheric pressure is 343.2 m/s whereas in water it is over 4 times as fast at 1,484 m/s. Secondly, the motion of the medium itself can affect the propagating wave. Any movement may cause an increase or decrease of the speed of the sound wave depending on the direction of the movement. This is best imagined by thinking about a sound wave moving through wind. The speed of propagation will increase if the sound wave moves in the same direction as the wind and will decrease if the sound and wind are moving in opposite directions. Finally, the viscosity of the medium also affects the propagation (Figure 5). For most media, such as air or water, this is a negligible effect, but for medium of greater viscosity a process called attenuation can occur. Attenuation is the gradual loss of intensity as the wave propagates, causing the sound to become quieter at a faster rate. Mathematics of Sinusoidal Waves To describe the motion of a wave, physicists use the concept of a wave function which describes the shape of the wave in space at any given time. Since there are many different types of waves, there are many different wave functions. The most basic of these wave functions are sinusoidal waves, which refers to a periodic wave or a wave with repetitive motion. To start developing the wave function we must first define a few inherent properties of the wave, assign them a symbol and give their SI units. These are: * Wave speed ( v ): The speed of the wave's propagation, given the units meters per second. * Amplitude ( A ): The maximum magnitude of the displacement from equilibrium, given in meters. * Period ( T ): The time for one full wave cycle to occur in seconds. This can be thought of as the time between two pulses, or from crest to crest / trough to trough. * Frequency ( f ): The number of cycles in a unit of time, given the unit hertz (Hz) which is equivalent to one cycle per second. * Angular frequency ( \omega ): This is 2\pi times the frequency with units of radians per second. * Wavelength ( \lambda ): The distance between any two points at corresponding positions on the successive repetitions in the wave, for example, from one crest or trough to the next, with units of meters. *Wave number ( k ): This is a useful quantity also known as the propagation constant which is defined as 2\pi divided by the wavelength, giving the units as radians per meter. Now we have the basic quantities defined, we can start building relations between them. From the definitions of period T and frequency f , we can see that each is the reciprocal of each other, giving the relationship: f=\dfrac{1}{T} Similarly, from the definition of angular frequency we can find the relationship: \omega=2\pi f=\dfrac{2\pi}{T} The wave pattern travels with constant speed v and advances a distance of one wavelength in a time interval of one period T so the wave speed is given by: v=\dfrac{\lambda}{T}=f\lambda Now we have relations between the basic properties of the wave, we can start constructing a wave function for a sinusoidal wave. Lets consider the example of a sinusoidal wave travelling from left to right (on the x-axis) along a string. Each particle in the string oscillates with the same amplitude and frequency, but the oscillations of particles at different points along the string are not in step with each other. For any two particle neighbours, the motion of the particle on the right lags behind the motion of the particle on the left by an amount proportional to the distance between them (Figure 6). Hence, the motion of various points on the string are out of step with each other by varying fractions of a cycle. These are called phase differences and it is said that the phase of the motion is different for different points along the string. First, lets consider a single point on the string. To mathematically describe its cyclic motion in the y-direction (perpendicular to the direction of wave propagation or x-direction) we use trigonometric functions. Lets say the particle starts at maximum amplitude and oscillates around its equilibrium point (y=0) over time. This can be described by a cosine function, which similarly starts at a maximum amplitude and oscillates around 0. Since the amplitude of a cosine function is 1, multiplying it by the wave amplitude A gives us a cyclic function which starts at an amplitude A and oscillates around a zero point. This is given by: y(t)=Acos(\omega t)=Acos(2\pi ft) This equation describes a particle that oscillates up and down in simple harmonic motion with amplitude A , frequency f and angular frequency \omega=2\pi f . For a description of simple harmonic motion, see the relevant section below. Now lets consider how the wave itself propagates along the string. If the wave disturbance travels from x=0 to some point x to the right of the origin then the amount of time this takes is given by x/v , where v is the wave speed. So the motion of point x at time t is the same as the motion of point x=0 at the earlier time t-x/v . Therefore we can find the displacement of point x at time t by simply replacing t in the above equation by (t-x/v) which gives us the following expression for the wave function: y(x,t)=Acos\bigg\bigg(t-\dfrac{x}{v}\bigg)\bigg Since cos(-\theta)=cos(\theta) , we can rewrite the wave function as: y(x,t)=Acos\bigg\bigg(\dfrac{x}{v}-t\bigg)\bigg=Acos\biggf \bigg(\dfrac{x}{v}-t\bigg)\bigg Now y(x,t) describes a sinusoidal wave moving in the + x -direction as is a function of both location x of the point and the time t . This wave function has several different but useful forms. For example we can express it in term of period T=1/f and the wavelength \lambda =v/f : y(x,t)=Acos\bigg\bigg(\dfrac{x}{\lambda}-\dfrac{t}{T}\bigg)\bigg To obtain the wave function in its most simplistic form, we use the wave number k=2\pi /\lambda and a formula for angular frequency \omega which is found by substituting \lambda =2\pi /k and f=\omega /2\pi into the wavelength-frequency relationship v=f\lambda giving \omega =vk . We can then rewrite the wave function as: y(x,t)=Acos(kx-\omega t) Since there are many forms of this wave function, which one you choose to use in any specific situation is a matter of convenience. Mathematics of Sound Waves Sound waves can be described in terms of variations in pressure at various points in the medium it travels through. For a sinusoidal wave in air, the pressure fluctuates above and below atmospheric pressure in a sinusoidal variation with the same frequency as the motion of the air particles. We describe this by p(x,t) which is the instantaneous pressure fluctuation in a sound wave at any point x at time t . This is the amount by which the pressure differs from the normal atmospheric pressure. The pressure fluctuation is related to the wave function for displacement y(x,t) by: p(x,t)=-B\dfrac{\partial y(x,t)}{\partial x} Where B is defined as the Bulk modulus. This is a property of the medium it travels through, and it measures its' resistance to uniform compression, measured in pascals (Pa). For example, the bulk modulus for air is 1.01\times 10^{5} Pa . If we use the wave function obtained in the Mathematics of Waves section, y(x,t)=Acos(kx-\omega t) , and evaluate \partial y(x,t)/\partial x using differentiation we find: p(x,t)=BkAsin(kx-\omega t) This now describes a sinusoidal pressure wave travelling through a medium. Now that the mathematics of waves has been introduced and expanded upon, we shall move onto the necessary concepts behind resonance. To understand resonance, first an understanding of oscillatory motion is required, followed by driven oscillations leading to resonance in systems. Simple Harmonic Motion (SHM) SHM is periodic motion where the restoring force, the force acting on the oscillating object in the direction of equilibrium, is proportional to the distance that the object is away from the equilibrium position. For example, this means that as you pull a mass on a spring, you are moving the mass away from its' equilibrium position, and as this distance increases, the force increases. You can typically feel this as the spring gets increasingly difficult to stretch, a larger force is opposing your movement. This allows us to move onto the bridge between SHM and Resonance, which is considering a system which is being driven, a driven harmonic oscillator. We shall neglect the maths behind these further concepts, and instead focus on what they are, and what causes them. An oscillator which has no external forces applied is a free oscillator. On the other hand, as will be discussed further in the section below, an oscillator which has an external force applied, whether this be continuous or change with time, is a forced oscillator. A mass on a spring has a natural frequency: f=\frac{1}{2\pi}\sqrt{\frac{k}{m}} Where k is the spring constant of the spring, and m is the mass. And for a pendulum, its' natural frequency is given by: f=\frac{1}{2\pi}\sqrt{\frac{g}{l}} Where g is the gravitational field strength, and l is the length of the pendulum. Driven Harmonic Oscillator A forced oscillator can be a driven harmonic oscillator if the force applied changes with time. If the force applied is continuous, doesn't change with time, then the oscillating system will eventually find a new equilibrium. Other than the position of equilibrium changing, a constant force does not alter the oscillations of a free system. A driven system is best explained through the analogy of a swing. A swing can oscillate like a pendulum, but when someone else pushes the swing, it becomes a driven harmonic oscillator. Depending on the timing and the force applied, the swing can do a range of things. If the swing is pushed in time with the oscillations, then its' amplitude will increase above what it could if it were a free oscillation. This is because the force being applied is time dependent, it is repetitive. Someone holding out their hands towards the swing but not moving them will not increase the swings amplitude. Resonance For the resonance tube experiment, it is necessary to understand the basic concepts of resonance that arise from certain systems. First of all, resonance occurs when there is a force driving an oscillating system, such as pushing a swing. All systems have a natural frequency which they oscillate at, and as the driving forces' frequency approaches this natural frequency, the amplitude of the systems oscillations increase, as shown in Figure 7. When the driving force oscillates at this natural frequency, it causes resonance to occur. This means that the oscillation amplitude is increased, and sometimes this can be a very dramatic increase in amplitude. Resonance can cause otherwise stable bridges being driven by wind at a resonant frequency to swing and sometimes even collapse. Now imagine the system from before with the swing. This swing has a natural frequency when it is a free oscillator. Imagine pushing it in time with this natural frequency, which is what the majority of people do without thinking why. If a force is applied at the same frequency as the natural frequency, the swings' amplitude can drastically increase. When the driven oscillating force is applied at the natural frequency, resonance occurs. The resonance taking place in the Resonance tube experiment uses the knowledge of waves and resonance together and is explained further in the experiment section of this wiki. However, it was necessary to delve into the meaning of resonance before performing an experiment which relies upon it.